I now accepted the challenge and prepared an extension to listen for all notifications, which works fine in MIT App Inventor, for details see here App Inventor Extensions: Notification Listener | Pura Vida Apps
Notification Listener Extension to listen to all notifications of your device. Packagename, title and text of all notifications will be stored in TinyDB. Also of course works if your app is not running and also survives a reboot of the device.
Minimum API level is 21 (Android 5).
However I’m using an AndroidX library and therefore it will not run in Kodular as it is. But I additionally provided the source code of the extension, so feel free to adjust it to your needs to get it running in Kodular. Please respect my license if you like to do this. You only have to replace one library or just remove the Local Broadcast Manager assuming the services annotation already works in the Kodular Extension IDE. You also can get an idea what needs to be done to get a service running in an extension.
Taifun
PS: I don’t think, that a lot of users will like to use it, because of the following message, which needs to be granted by the user… you might want to convince me from the contrary… 
